TY - JOUR
T1 - A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation
AU - Li, Guopeng
AU - Li, Jiawei
AU - Tolomeo, Leonardo
N1 - Publisher Copyright:
© The Author(s) 2026.
PY - 2026
Y1 - 2026
N2 - We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.
AB - We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.
KW - Gibbs measure
KW - Logarithmic Sobolev inequalities
KW - Nonlinear Schrödinger equation
UR - https://www.scopus.com/pages/publications/105037472085
U2 - 10.1007/s10884-026-10509-y
DO - 10.1007/s10884-026-10509-y
M3 - Article
AN - SCOPUS:105037472085
SN - 1040-7294
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
ER -